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STEM.M.MATHEMATICS: National Council of Teachers of Mathematics (NCTM)
MATHEMATICS: National Council of Teachers of Mathematics (NCTM)
NCTM.1. Number and Operations 1.1. Understand numbers, ways of representing numbers, relationships among numbers, and number systems. 1.1.1. Work flexibly with fractions, decimals, and percents to solve problems.
1.1.2. Compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line.
1.1.3. Develop meaning for percents greater than 100 and less than 1.
1.1.4. Understand and use ratios and proportions to represent quantitative relationships.
1.1.5. Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation.
1.1.6. Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
1.1.7. Develop meaning for integers and represent and compare quantities with them.
1.2. Understand meanings of operations and how they relate to one another. 1.2.1. Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers.
1.2.2. Use the associative and commutative properties of addition and multiplication and the distributive property of multiplication over addition to simplify computations with integers, fractions, and decimals.
1.2.3. Understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.
1.3. Compute fluently and make reasonable estimates. 1.3.1. Select appropriate methods and tools for computing with fractions and decimals from among mental computation, estimation, calculators or computers, and paper and pencil, depending on the situation, and apply the selected methods.
1.3.2. Develop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their use.
1.3.4. Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
NCTM.11. Grade 8 Curriculum Focal Points 11.1. Algebra: Analyzing and representing linear functions and solving linear equations and systems of linear equations 11.1.1. Students use linear functions, linear equations, and systems of linear equations to represent, analyze, and solve a variety of problems. They recognize a proportion (y/x = k, or y = kx) as a special case of a linear equation of the form y = mx + b, understanding that the constant of proportionality (k) is the slope and the resulting graph is a line through the origin. Students understand that the slope (m) of a line is a constant rate of change, so if the input, or x-coordinate, changes by a specific amount, a, the output, or y-coordinate, changes by the amount ma. Students translate among verbal, tabular, graphical, and algebraic representations of functions (recognizing that tabular and graphical representations are usually only partial representations), and they describe how such aspects of a function as slope and y-intercept appear in different representations. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines that intersect, are parallel, or are the same line, in the plane. Students use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems. Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
11.2. Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle 11.2.1. Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.
11.3. Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets 11.3.1. Students use descriptive statistics, including mean, median, and range, to summarize and compare data sets, and they organize and display data to pose and answer questions. They compare the information provided by the mean and the median and investigate the different effects that changes in data values have on these measures of center. They understand that a measure of center alone does not thoroughly describe a data set because very different data sets can share the same measure of center. Students select the mean or the median as the appropriate measure of center for a given purpose.
NCTM.12. Connections to the Grade 8 Focal Points 12.1. Algebra: Students encounter some nonlinear functions (such as the inverse proportions that they studied in grade 7 as well as basic quadratic and exponential functions) whose rates of change contrast with the constant rate of change of linear functions. They view arithmetic sequences, including those arising from patterns or problems, as linear functions whose inputs are counting numbers. They apply ideas about linear functions to solve problems involving rates such as motion at a constant speed. Quiz, Flash Cards, Worksheet, Game & Study Guide Functions Quiz, Flash Cards, Worksheet, Game & Study Guide Sequences
12.2. Geometry: Given a line in a coordinate plane, students understand that all 'slope triangles' - triangles created by a vertical 'rise' line segment (showing the change in y), a horizontal 'run' line segment (showing the change in x), and a segment of the line itself - are similar. They also understand the relationship of these similar triangles to the constant slope of a line.
12.3. Data Analysis: Building on their work in previous grades to organize and display data to pose and answer questions, students now see numerical data as an aggregate, which they can often summarize with one or several numbers. In addition to the median, students determine the 25th and 75th percentiles (1st and 3rd quartiles) to obtain information about the spread of data. They may use box-and-whisker plots to convey this information. Students make scatterplots to display bivariate data, and they informally estimate lines of best fit to make and test conjectures.
12.4. Number and Operations: Students use exponents and scientific notation to describe very large and very small numbers. They use square roots when they apply the Pythagorean Theorem.
NCTM.2. Algebra 2.1. Understand patterns, relations, and functions. 2.1.1. Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. Quiz, Flash Cards, Worksheet, Game & Study Guide Sequences
2.2. Represent and analyze mathematical situations and structures using algebraic symbols. 2.2.1. Develop an initial conceptual understanding of different uses of variables.
2.2.2. Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.
2.2.3. Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships.
2.2.4. Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
2.3. Use mathematical models to represent and understand quantitative relationships. 2.3.1. Model and solve contextualized problems using various representations, such as graphs, tables, and equations.
2.4. Analyze change in various contexts. 2.4.1. Use graphs to analyze the nature of changes in quantities in linear relationships.
NCTM.3. Geometry 3.1. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. 3.1.1. Precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties.
3.1.2. Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
3.1.3. Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
3.2. Specify locations and describe spatial relationships using coordinate geometry and other representational systems. 3.2.1. Use coordinate geometry to represent and examine the properties of geometric shapes.
3.2.2. Use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
3.3. Apply transformations and use symmetry to analyze mathematical situations. 3.3.1. Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
3.3.2. Examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
3.4. Use visualization, spatial reasoning, and geometric modeling to solve problems. 3.4.4. Use geometric models to represent and explain numerical and algebraic relationships.
NCTM.4. Measurement 4.1. Understand measurable attributes of objects and the units, systems, and processes of measurement. 4.1.3. Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
4.2. Apply appropriate techniques, tools, and formulas to determine measurements. 4.2.2. Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.
4.2.3. Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.
4.2.4. Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders.
4.2.5. Solve problems involving scale factors, using ratio and proportion.
4.2.6. Solve simple problems involving rates and derived measurements for such attributes as velocity and density.
NCTM.5. Data Analysis and Probability 5.1. Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. 5.1.1. Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.
5.1.2. Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.
5.2. Select and use appropriate statistical methods to analyze data. 5.2.1. Find, use, and interpret measures of center and spread, including mean and interquartile range.
5.2.2. Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.
5.3. Develop and evaluate inferences and predictions that are based on data. 5.3.2. Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.
5.3.3. Use conjectures to formulate new questions and plan new studies to answer them.
5.4. Understand and apply basic concepts of probability 5.4.2. Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations.
5.4.3. Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
NCTM.6. Problem Solving 6.1. Build new mathematical knowledge through problem solving.
6.2. Solve problems that arise in mathematics and in other contexts.
6.3. Apply and adapt a variety of appropriate strategies to solve problems.
NCTM.7. Reasoning and Proof 7.1. Recognize reasoning and proof as fundamental aspects of mathematics.
7.2. Make and investigate mathematical conjectures.
7.3. Develop and evaluate mathematical arguments and proofs.
7.4. Select and use various types of reasoning and methods of proof.
NCTM.9. Connections 9.2. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.